HOMEWORK (WEEK 1):
Questions on Present Value Computations:
You just purchased a new SUV. You financed your SUV with a loan consisting of 60 monthly repayments, each
occurring at the end of the month. The first re-payment (after one month) equals £325, but the monthly re-
payments grow at a constant rate of 0.25%. The effective monthly interest rate is 0.5%.
1. What is the value of the loan at the time you purchased the SUV?
Solution: Note that this stream of cash flows constitutes a growing annuity over 60 periods. It can hence be
valued using: PV(C growing at g until N) = C/(r-g)*[1-((1+g)/(1+r))^N].
Plugging numbers into the formula: 325/(0.005-0.0025)*[1-(1.0025/1.005)^60] = 18045.25.
Two years have passed, and you made the first 12 re-payments on time.
2. What is the value of the loan after one year has passed?
Solution: This is a bit tricky. After twelve months have passed, the first of the 48 remaining cash flows is
equal to 325*(1+0.0025)^12 = 334.8852 (t=13), and this cash flow will occur after one month. The value of
the remaining cash flows can therefore be again calculated from the formula for a growing annuity, this time
however with the first cash flow equal to 334.8852 and the remaining number of periods equal to 48, i.e.,
334.8852/(0.005-0.0025)*[1-(1.0025/1.005)^48] = 15094.20.
The difference between the initial value of the loan and its value after 12 months is equal to the total value
of the loan that you have already repaid.
3. What is the difference between the initial value of the loan and the value of the loan after one year
has passed (i.e., how much of the loan have you already repaid)?
Solution: 18045.25 - 15094.20 = 2951.05.
4. Also compute the total sum of re-payments over the first twelve months (i.e., 325+325(1*.0025)+...).
Solution: 325 + 325*(1.0025) + 325*(1.0025)^2 + 325*(1.0025)^3 + 325*(1.0025)^4 + 325*(1.0025)^5 + 325
(1.0025)^6 +325*(1.0025)^7 + 325*(1.0025)^8 + 325*(1.0025)^9 + 325*(1.0025)^10 + 325*(1.0025)^11 =
3954.074 (NOTE: One could also calculate this using the formula for a growing annuity over 12 periods with
the effective interest rate set equal to zero!).
You should notice that this exceeds the amount of the loan that you already repaid (the answer to question
3). In fact, the difference is the total interest you paid in the first 12 months.
5. How much did you already repay in interest?
Solution: 3954.074 - 2951.05 = 1003.24.
You are unhappy with the terms of the loan contract. In fact, you would like to borrow the same amount of
money as with the original loan (the answer to question 1), but you want to repay the loan in 120 monthly
re-payments – and not 60. Also, you would like the monthly payments to remain constant over time.
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Questions on Present Value Computations:
You just purchased a new SUV. You financed your SUV with a loan consisting of 60 monthly repayments, each
occurring at the end of the month. The first re-payment (after one month) equals £325, but the monthly re-
payments grow at a constant rate of 0.25%. The effective monthly interest rate is 0.5%.
1. What is the value of the loan at the time you purchased the SUV?
Solution: Note that this stream of cash flows constitutes a growing annuity over 60 periods. It can hence be
valued using: PV(C growing at g until N) = C/(r-g)*[1-((1+g)/(1+r))^N].
Plugging numbers into the formula: 325/(0.005-0.0025)*[1-(1.0025/1.005)^60] = 18045.25.
Two years have passed, and you made the first 12 re-payments on time.
2. What is the value of the loan after one year has passed?
Solution: This is a bit tricky. After twelve months have passed, the first of the 48 remaining cash flows is
equal to 325*(1+0.0025)^12 = 334.8852 (t=13), and this cash flow will occur after one month. The value of
the remaining cash flows can therefore be again calculated from the formula for a growing annuity, this time
however with the first cash flow equal to 334.8852 and the remaining number of periods equal to 48, i.e.,
334.8852/(0.005-0.0025)*[1-(1.0025/1.005)^48] = 15094.20.
The difference between the initial value of the loan and its value after 12 months is equal to the total value
of the loan that you have already repaid.
3. What is the difference between the initial value of the loan and the value of the loan after one year
has passed (i.e., how much of the loan have you already repaid)?
Solution: 18045.25 - 15094.20 = 2951.05.
4. Also compute the total sum of re-payments over the first twelve months (i.e., 325+325(1*.0025)+...).
Solution: 325 + 325*(1.0025) + 325*(1.0025)^2 + 325*(1.0025)^3 + 325*(1.0025)^4 + 325*(1.0025)^5 + 325
(1.0025)^6 +325*(1.0025)^7 + 325*(1.0025)^8 + 325*(1.0025)^9 + 325*(1.0025)^10 + 325*(1.0025)^11 =
3954.074 (NOTE: One could also calculate this using the formula for a growing annuity over 12 periods with
the effective interest rate set equal to zero!).
You should notice that this exceeds the amount of the loan that you already repaid (the answer to question
3). In fact, the difference is the total interest you paid in the first 12 months.
5. How much did you already repay in interest?
Solution: 3954.074 - 2951.05 = 1003.24.
You are unhappy with the terms of the loan contract. In fact, you would like to borrow the same amount of
money as with the original loan (the answer to question 1), but you want to repay the loan in 120 monthly
re-payments – and not 60. Also, you would like the monthly payments to remain constant over time.
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